Influence of Charge Regulation on the Performance of Shock Electrodialysis

In order to understand the ion transport in a continuous cross-flow shock electrodialysis process better, numerous theoretical studies have been carried out. One major assumption involved in these models has been that of a constant surface charge. In this work, we considered the influence of charge regulation, caused by changes in salt concentration, on the performance of a shock electrodialysis cell. Our results show that, by including charge regulation, much higher potentials need to be applied to reach the same degree of desalination, compared to the constant surface charge model. Furthermore, we found that operating at higher potentials could lead to substantial Joule heating and therefore temperature increases. Although somewhat lower potentials were required in the nonisothermal case versus the isothermal case with charge regulation, the required energy input for desalination is still much higher than the thermodynamic minimum. This works highlights the important role charge regulation can play in a shock electrodialysis process.

S1 Detailed information on other temperature effects S1.1 Buoyancy effect Free convection arises when are density differences along the Earth's gravitational field.
The density of water decreases with temperature and therefore temperature gradients could lead to natural convection. Since it is established in section 6.1 that there is only a large temperature gradient along the length of the setup. Only when the setup is positioned as indicated on the right of Figure S1 this could potentially lead to natural convection. To evaluate the influence of the free convection in that particular case with respect to the forced convective fluid flow, the Richardson number is evaluated, which gives the ratio between forced convection and natural convection as ratio between the Grashof and the square Reynolds numbers The thermal Grashof number is defined as: with g acceleration due to gravity,β the thermal expansion coefficient of water and ν the kinematic viscosity of water. By filling in the approximate overall temperature difference between the inlet and outlet of the non-isothermal charge regulation model at 90 V, the Grashof number is approximately equal to 930.
The Reynolds number is given by: Under the same circumstances as for the previous calculation of the Grashof number, the Reynolds number is equal to approximately 0.1. This results in a Richardson number of 93000.
This means that when the shock ED is oriented as indicated by the right of Figure S1, the free convection will probably completely dominate the pressure induced forced convection.
Therefore, with a temperature difference the water should be pumped through at a higher pressure to overcome the natural convective flows, which will increase the salinity of the depleted stream at similar applied potentials. Therefore, in practice it is highly undesirable to allow natural convection to occur in the shock ED process. However, based on the geometry, such a setup will most likely be positioned like the left of Figure S1 and in that case free convection will not play such a distinct role in the hydrodynamics.

S1.2 Soret Effect
. The Soret effect, also known as thermophoresis, results in enhanced ion transport, when there are considerable temperature gradients present in a system. The Soret flux for the cations can be described by: where the Q * is the heat of transport which is for Na + equal to 3.46 kJ/mol as given by By comparing the flux expression to the total flux towards the electrode boundary of the non-isothermal charge regulation model, it was calculated that the Soret flux will be at least 1000 times smaller, outside the depletion zone compared to the total ion flux. While inside the depletion zone this ratio was even larger and the Soret flux would be approximately a million times smaller compared to the total flux. This can be attributed to the lack of a large temperature gradient, as discussed in previous section, that drives the ions. Therefore it is probably safe to conclude that the effect of thermophoresis will be negligible in shock ED simulations.

S1.3 Effect of Temperature on the Dielectric Constant
The effect of temperature on the relative dielectric constant of water is discussed. Since in our description of shock ED the electroneutrality condition is used, the dielectric constant only affects the Debye length (Debye parameter), which in turn affects the surface charge density of the porous medium. To describe the temperature dependence, the following empirically determined equation was used. 2 where r is the relative dielectric constant and T is the temperature in degree C. Note that this empirical relation is strictly only valid for water without any salt. However, the dielectric constant is not very sensitive to the salt concentrations and only for water such as sea water, it will change the dielectric constant 3 significantly. Since in our work we only

S2 Electroosmotic vortices in porous media
Recently it was found that electroosmotic vortices may form in porous materials during shock ED, due to different local pressures within the connected pores. This could enhance the ion S5 transport in shock ED. 5 However, in the continuum model presented here, the electroosmotic flow is largely counteracted by the resulting pressure gradient at the bottom wall, which means that net ion transport was not strongly affected by convective flows.
To investigate the effect of electroosmotic vortices with a continuum model, we included an enhanced diffusion coefficient description based on Taylor-Aris kind of dispersion as described by Licon Bernal et al. 6 The enhanced diffusion coefficient due to electroosmotic vortices is described by where h is originally the microchannel size and here it is assumed to be equal to the average pore size (h p ), u EO is the electroosmotic velocity (in the y-direction) To ensure the effect of electroosmotic vortices is only applied in the direction of the electroosmotic flow, we added the diffusion coefficients as a so-called diagonal matrix, with the usual diffusion coefficient in x-direction and the enhanced diffusion coefficient in the y-direction.
When evaluating the ratio of the enhanced diffusion coefficient over the usual diffusion coefficient, calculated by takeing the average over the bottom electrode boundary, It can be seen that at these potentials, the enhancement is about 13 to 16 percent compared to the temperature corrected diffusion coefficient (see Table S2). While this does not seem to be a very large difference, the relative difference in the outlet concentrations ranges from about 15 to 70 percent at a potential of 35 and 55 V respectively, when comparing the electroosmotic vortices case with the CR case. This difference is probably due to the fact that for increasing potential, the depletion layer grows in size, which leads to a larger region where the diffusion coefficient is higher compared to the bulk value. However, one should realize that although the relative differences are quite large, the absolute numbers are at that point already quite small.

S6
To conclude, when the effect of electroosmotic vortices on the ion transport is included, the ion transport is slightly enhanced, which means that it could be a valuable addition to describe the shock ED process better. When put in perspective the influence of chargeregulation (variable charge density) was much larger relatively. However, since the correlation used was developed for electroosmotic vortices enhancing diffusion in a single microchannel, it is possible that these effects could play a larger role. This analysis shows that the effect on ion transport of electroosmotic vortices might be limited in continuum simulations. In a more realistic shock ED setup that includes transitions from free fluid to the porous medium (region 1 to 2 in Figure 3) and vice versa (region 2 to 3 in Figure 3), as has been done by Tian et al. 4,7 In this work, it was shown that electroosmotic flows near these transitions also result large electroosmotic vortices, which partially mixes the feed salt solution with the depletion layer. These vortices will results in an increase the energy consumption of Shock ED processes compared to the energy consumption considered in our simulation cases.

S3 Effect of fixed pH assumption
Throughout this work, we neglected the effect of the formation of protons and hydroxides due to the water equilibrium or electrode reactions. Naturally, the current efficiency will drop due to the presence of these ions, which seems to lead to deviations between experimental and our simulation results. However, since in reality the protons will be transported through S7 the ion exchange membrane, it is expected that within the depleted layer the pH will remain rather constant, which is also suggested by the 2D figures presented in the work of Tian et al. 4 This means that due to the lower current efficiency the driving force needs to be somewhat higher compared to the results presented in this work but the charge of the porous medium is not altered too much in the simulations. Therefore it is expected that this assumption will not have as much effect on the theoretical performance as the charge regulation and Joule heating have.

S8
In Figure S2, the desalination performance for 3 constant charge models with different charge densities is presented. In this figure it can be seen that, especially at the highest applied potentials, for a certain depleted concentration, the charge density determines proportionally the potential that needs to be applied to reach a certain degree of desalination. This shows the importance of having a porous medium that has higher charge densities in similar salt concentrations compared to silica to improve the efficiency of the shock ED process to some extent.